nLab
bifibration

Contents

Definition

A bifibration of categories is a functor

EB E \to B

that is both a Grothendieck fibration as well as an opfibration.

Therefore every morphism f:b 1b 2f : b_1 \to b_2 in a bifibration has both a push-forward f *:E b 1E b 2f_* : E_{b_1} \to E_{b_2} as well as a pullback f *:E b 2E b 1f^* : E_{b_2} \to E_{b_1}.

Examples

Relation to monadic descent

Ordinary Grothendieck fibrations correspond to pseudofunctors to Cat, by the Grothendieck construction and hence to prestacks. For these one may consider descent.

If the fibration is even a bifibration, there is a particularly elegant algebraic way to encode its descent properties; this is monadic descent. The Benabou–Roubaud theorem characterizes descent properties for bifibrations.

Relation to distributive laws

Fibrations and opfibrations on a category CC (or more generally an object of a suitable 2-category) are the algebras for a pair of pseudomonads. If CC has pullbacks, there is a pseudo-distributive law between these pseudomonads, whose joint algebras are the bifibrations satisfying the Beck-Chevalley condition; see (von Glehn).

Remark

A bifibration F:EBF:E\to B such that F op:E opBF^{op}:E^{op}\to B is a bifibration as well is called a trifibration (cf. Pavlović 1990, p.315).

Bifibration of bicategories

The basic theory of 2-fibrations, fibrations of bicategories, is developed in (Buckley). One (unpublished) proposal by Buckley and Shulman, for a “2-bifibration”, a bifibration of bicategories, is a functor of bicategories that is both a 2-fibration and a 2-opfibration. This should correspond to a pseudofunctor into 2Cat adj2 Cat_{adj}, just as for 1-categories. There is a difference from this case, however, in that although a 2-fibration (corresponding to a “doubly contravariant” pseudofunctor B coop2CatB^{coop} \to 2Cat) has both cartesian lifts of 2-cells and 1-cells, a 2-opfibration (corresponding to a totally covariant pseudofunctor B2CatB\to 2Cat) has opcartesian lifts of 1-cells but still cartesian lifts of 2-cells.

References

Last revised on May 5, 2018 at 03:09:01. See the history of this page for a list of all contributions to it.